Quantum Wavepackets, Bohemian Potential and Interaction Probability

Quantum Wavepackets, Bohemian Potential and Interaction Probability Francesco R. Ruggeri Hanwell, N.B. Dec. 19, 2025 Traditional quantum mechanics calculates a probability in space through W*(x)W(x) for a time-independent wavefunction W(x) = Sum over p a(p)exp(ip dot r). We argue that this probability is created at the interaction region and indicates a probability for outcomes. In particular, given a 2-slit problem, one calculates W(r) = exp( i p dot r1) + exp(i p dot r2), where r1 and r2 are vectors from the center of each slit to the same point on a screen far away. W(r)*W(r) then represents the probability for particles to arrive at various points on the screen. This may be realized by sending many particles/photons through the 2-slit apparatus, one at a time. Given that exp(ip dot r) is an interaction probability (we argue), its relevance holds in the interaction region which is presumably of about a wavelength long or so. We argue that exp(i p dot r) exp(-i p dot r) = 1 does not mean that one should interpret the particle/photon as being possibly present at any r in space. The particle still has a center-of-mass which follows x=vt for a free particle. The only issue is that exp(ipx) implies that the particle does not interact at its center-of-mass point necessarily, but can interact with probability exp(ipx) at different points within .5 hbar/p of the center of mass on either side. As a result, we argue that there is no issue concerning localization of either a free particle or photon. The object is localized and moving at x=vt and it has a precise p which may render impulse hits at points about the center-of-mass point in keeping with exp(ipx). As a result, Heisenberg’s uncertainty relation (standard deviation p) (standard deviation x) >= hbar/2 does not hold for exp(ipx) because standard deviation p =0. exp(ipx) applies to a free particle which is not interacting. When the particle interacts (which is the way it is measured), then exp(ipx) becomes Sum over p a(p)exp(i p dot r). In such a case, a standard deviation of p and x arise. The Heisenberg uncertainty principle holds for interactions which are the only way the particle can be measured and so applies to W(x) and not a single exp(ipx). Given the above ideas, we suggest there is no reason to create a free particle wavepacket. Such a wavepacket seems to be created in the literature (1) in order to localize a particle in space and to ensure that Heisenberg’s uncertainty relationship is enforced. We have argued that there is no reason for this relationship if the particle is not interacting. As for localization, the free particle is localized by its center-of-mass in x=vt. It is just that one is not measuring this point. Furthermore, the increase in a Gaussian wavepacket’s x standard deviation in time is unphysical. Given that there is no reason for a wavepacket, we then question the notion of the wavepacket as being associated with a Bohmian potential which creates the wavepacket so to speak, i.e. -1/2m (d/dx dd/x A(x)) / A(x), where W(x,t) = A(x) exp(i S(x,t) ). Momentum Conserving Probability We have suggested in previous notes that one may introduce a probability for finding outcomes in Newtonian 2-body elastic scattering. Given the Newtonian nature of the situation, each particle has a precise p and the interaction occurs at a sharp x point mathematically. Physically, it occurs in some dx region, but if one tries to create a probability P(p) and it turns out this also depends on x, then P(p,x) is a math probability function which idealizes the interaction as occurring at a sharp x point. The same sharp x occurs for an n1-n2 index of refraction junction even though interactions do not occur at a point. This, we argue, is a math generalization which may be used in a probabilistic treatment. We suggest that given an initial e1,e2 energies and p1,p2 momentum vectors (in the x direction for simplicity), then any ei, ej and pi, pj which satisfy momentum and energy conservation have the same probability of occurring. This suggests: exp(i C1 E) and exp(i C2 p) with C1, C2 being unit fixing constant ((1)) For p=m1v1=m2v2, both particles satisfy a momentum conservation equation in one frame, but not in a boosted frame and so we argue that the probability must be Lorentz invariant and so must contain x,t in addition to E,p, i.e. exp(-iEt+ipx) ((2)) If one considers only momentum conservation, then one uses exp(ipx). “p” is a precise value and so is x. Momentum conservation is described by: exp(-ip3x)exp(-ip4x) exp(ip1x) exp(ip2x) with p1+p2=p3+p4 (x direction and exp(i p dot r) in general ((3)) As a result, exp(ipx) is a math probability function. Real interactions do not physically occur at a point, but mathematically one has the notion of a center-of-mass point and one may consider conservation of momentum at such an idealized point as in ((3)). In ((3)), all exp(ipx)s have the same x. As a result, one is able to obtain exp(ipx) through the above arguments and there is no issue of an uncertain p or an nonlocalized particle. exp(ipx) is associated with a dx = hbar/p which, we argue, means that for the particle at x,t, following x=vt, an impulse hit of p may probabilistically occur in a dx=hbar/p region about the x,t point exp(ipx) describes this interaction. exp(ipx)exp(-ipx) = 1 ((4)) simply shows that exp(ipx) is consistent with the treatment of each x point as having equal weight in x=vt. It does not mean that the particle may be anywhere along the x-axis. exp(ipx) is an interaction probability and an interaction occurs in an exp(ipx) reason. It is, however, possible for exp(ipx) to receive phase shifts, but these are relative to the hbar/p length, i.e. one may subtract 2*3.14 *n, where n is an integer ((5)) from any phase. We suggest that a free particle exp(ipx) is linked with a precise p and even x is sharp. The only issue is that one does not know at which x point within hbar/p a p impulse will occur. It does not occur necessarily at the center-of-mass point. Heisenberg Uncertainty Principle The Heisenberg uncertainty principle: (standard deviation p) (standard deviation x) >= hbar/2 ((6)) is applied in (1) to a free particle. A wavepacket is introduced so that ((6)) applies. ((6)) implies that a measurement has occurred, otherwise one could have a standard deviation value. A free particle, on the other hand, is not undergoing any interaction. We argue that one may write exp(ipx) ((7)) for a free particle such that standard deviation p =0 There is no contradiction with the Heisenberg uncertainty principle because, as noted. no interaction has occurred and so there is no measurement. exp(ipx) is then a math probability which may be used in probability calculations within an interaction region. For example, in a 2-slit case, one has: W(x) = exp(i p dot r1) + exp(i p dot r2) ((8)) where r1 is a vector from the center of slit 1 to a point on a screen and r2, from the center of slit 2 to the same point. The object of W(x) is to calculate W*(x)W(x) to see which x points at the screen (x is along the screen) are preferred. This is a probabilistic calculation, but for a given direction the particle or photon moves as x=vt (x=ct). W(x) As Describing an Interaction We have argued that exp(ipx) represents a free particle probability which does not satisfy Heisenberg’s uncertainty relation because no interaction occurs. If an interaction is present, then exp(ipx) is replaced with W(x) = sum over p a(p)exp(ipx). This represents the overall probability associated with the interaction and needs to be taken seriously. W(x) ((8)) yields a probability pattern in space for 2-slit interference. One does not need to consider whether the particle went through slit 1 or 2. It probabilistically interacted with both to produce a certain W*(x)W(x) scheme which then describes probabilities in space for finding the particle. Similarly, for an infinite well potential, W(x) = 0 outside the walls. This means that there is no probability to find a particle outside the wall, even though exp(ipx)s exist outside the wall. It is the overall probability which governs possibilities and so we argue W(x) must be taken seriously. It is W(x) which is linked to the Heisenberg uncertainty principle, we argue, because W(x) is associated with an interaction. Wavepackets A wavepacket is a linear combination of free particle exp(-iEt+ipx)’s i.e. W(x,t) = Sum over p a(p) exp(-iEt+px) where E=pp/2m in the nonrelativistic case ((9)) ((9)) is constructed so as to satisfy the Heisenberg uncertainty principle. Furthermore, W(x,t) localizes the free particle. A free particle, however, follows x=vt and exp(-iEt) and exp(ipx) describe interactions outside the x,t of the center-of-mass point within dt=hbar/E and dx= hbar/p. There is no reason for exp(-iEt+ipx) to be localized or to satisfy the Heisenberg uncertainty principle. We thus argue against the creation of a wavepacket. Furthermore, as is well-known (and also shown in (1)), a Gaussian wavepacket has its x standard deviation increase in time which is unphysical. Bohmian Potential In (1), a Bohmian potential -1/2m (d/dx dA/dx ) / A where W(x) = A(x) exp(i S(x,t)) is proposed in conjunction with the Gaussian wavepacket for a free particle. In other words, a Bohmian potential exists for a free particle because a wavepacket exists. If a wavepacket does not exist, as we argue, then there is no need for a Bohmian potential for a free particle. Conclusion In conclusion, we argue that one may arrive at the probability exp(-iEt+ipx) by considering Newtonian elastic 2-body scattering of free particles. The probability is a complex number with modulus 1 because each free particle carries the same real weight. The probability exists to ensure equal likelihood of any ei,ej, pi, pj (vectors) outcomes which conserve energy and momentum. x,t appear in exp(-iEt+ipx) because it must be Lorentz invariant. One has the notion of x=vt with a sharp center-of-mass and p and E are sharp values. Furthermore, the elastic collision occurs “mathematically” at a sharp x point, i.e. exp(-i p3x) exp(-i p4x) exp(ip1x)exp(ip2x) with p1+p2=p3+p4 (all momenta along the x-axis for simplicity). exp(ipx), however, implies a dx=hbar/p for a sharp p. This does not satisfy the Heisenberg uncertainty principle because (standard deviation p)=0, but we argue that this is fine. exp(ipx) represents a free particle which is not interacting. The Heisenberg uncertainty principle represents measured values (i.e. standard deviations) and so there must be an interaction occurring. In such a case, one must use W(x) = Sum over p a(p)exp(ipx) and not exp(ipx). Thus, for a free particle, we argue that exp(ipx) is fine. This implies then that one does not need a wavepacket for a free particle. In (1), a Gaussian wavepacket is introduced due to the requirement of the Heisenberg uncertainty relation. We argue that there is no such requirement for a free particle as it is not interacting and there is no p standard deviation. The Gaussian wavepacket in (1) is then used to justify a Bohmian potential -1/2 m (d/dx dA/dx) / A for a free particle with W(x) = A(x) exp(i S(x,t)). If a wavepacket description does not apply (and one may note the unphysical increasing x standard deviation of the packet), then one does not need a corresponding Bohmian potential for a free particle, we argue. References 1. Grande, P.L. and Fadanelli, R.C. and Vos, M. A Valsov-Bohm Approach to Quantum mechanics for statistical systems (2025) https://www.semanticscholar.org/reader/0fba7926dc5f1cfc9eedb995eea3a45da1a4b38e